(To be removed) Simulate univariate GARCH(P,Q) process with Gaussian innovations

As an alternative to ugarchsim, use the garch object to create conditional variance models. For more information, see Specify GARCH Models Using garch.

ugarchsim is removed. Use garch object to create conditional variance models and the simulate function to generate Monte Carlo simulations from conditional variance models. For more information, see Simulate Conditional Variance Model.


[U, H] = ugarchsim(Kappa, Alpha, Beta, NumSamples)



Scalar constant term [[KAPPA]] of the GARCH process.


P-by-1 vector of coefficients, where P is the number of lags of the conditional variance included in the GARCH process. Alpha can be an empty matrix, in which case P is assumed 0; when P = 0, a GARCH(0,Q) process is actually an ARCH(Q) process.


Q-by-1 vector of coefficients, where Q is the number of lags of the squared innovations included in the GARCH process.


Positive, scalar integer indicating the number of samples of the innovations U and conditional variance H (see below) to simulate.


[U, H] = ugarchsim(Kappa, Alpha, Beta, NumSamples) simulates a univariate GARCH(P,Q) process with Gaussian innovations.

U is a number of samples (NUMSAMPLES)-by-1 vector of innovations (ɛt), representing a mean-zero, discrete-time stochastic process. The innovations time series U is designed to follow the GARCH(P,Q) process specified by the inputs Kappa, Alpha, and Beta.

H is a NUMSAMPLES-by-1 vector of the conditional variances ( ) corresponding to the innovations vector U. Note that U and H are the same length, and form a "matching" pair of vectors. As shown in the following equation, σt2 (that is, H(t)) represents the time series inferred from the innovations time series {ɛt} (that is, U).

The time-conditional variance, σt2, of a GARCH(P,Q) process is modeled as


where α represents the argument Alpha, β represents Beta, and the GARCH(P,Q) coefficients {Κ, α, β} are subject to the following constraints.


Note that U is a vector of residuals or innovations (ɛt) of an econometric model, representing a mean-zero, discrete-time stochastic process.

Although σt2 is generated using the equation above, ɛt and σt2 are related as


where {υt} is an independent, identically distributed (iid) sequence ~ N(0,1).

The output vectors U and H are designed to be steady-state sequences in which transients have arbitrarily small effect. The (arbitrary) metric used by ugarchsim strips the first N samples of U and H such that the sum of the GARCH coefficients, excluding Kappa, raised to the Nth power, does not exceed 0.01.

       0.01 = (sum(Alpha) + sum(Beta))^N


       N = log(0.01)/log((sum(Alpha) + sum(Beta)))

    Note   The Econometrics Toolbox™ software provides a comprehensive and integrated computing environment for the analysis of volatility in time series. For information see the Econometrics Toolbox documentation or the financial products Web page at


This example simulates a GARCH(P,Q) process with P = 2 and Q = 1.

% Set the random number generator seed for reproducability.


% Set the simulation parameters of GARCH(P,Q) = GARCH(2,1) process.

Kappa = 0.25;      %a positive scalar.
Alpha = [0.2 0.1]'; %a column vector of nonnegative numbers (P = 2).
Beta = 0.4;        % Q = 1.
NumSamples = 500;  % number of samples to simulate.

% Now simulate the process.

[U , H] = ugarchsim(Kappa, Alpha, Beta, NumSamples);

% Estimate the process parameters.

P = 2;    % Model order P (P = length of Alpha).
Q = 1;    % Model order Q (Q = length of Beta).
[k, a, b] = ugarch(U , P , Q);
disp(' ')
disp(' Estimated Coefficients:')
disp(' -----------------------')
disp([k; a; b])
disp(' ')

% Forecast the conditional variance using the estimated   
% coefficients.

NumPeriods = 10;    % Forecast out to 10 periods.
[VarianceForecast, H1] = ugarchpred(U, k, a, b, NumPeriods);
disp(' Variance Forecasts:')
disp(' ------------------')
disp(' ')
Error using ugarchsim (line 71)
UGARCHSIM has been removed. Use the GARCH model SIMULATE function in Econometrics Toolbox instead.

Related Examples


James D. Hamilton, Time Series Analysis, Princeton University Press, 1994

See Also


Introduced before R2006a

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